Finite Group Databasebeta

← #11787568

Subgroups — #11787568

Order 1152.

1122 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3744C211yesmaximal, characteristic
2C2×C4×#634C211yesmaximal, characteristic, complemented
3C2×#63648C211yesmaximal, characteristic
4C2×#63643C211yesmaximal, characteristic, complemented
5#10592638C211yesmaximal, characteristic
6C32×#336C211yesmaximal, characteristic, complemented
7#10589417C211yesmaximal, characteristic, complemented
8#68686—34nomaximal
9C22×C4⋊(C3⋊D3)C2211yescharacteristic
10C22×#634C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C2×#3752—21no—
15C2×#3752—21no—
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×#3750—21no—
19C2×C4×C3⋊Dic3—21no—
20C2×#3744—22no—
21C2×#3744—22no—
22C2×#3744C414yescomplemented
23C4×#634—22no—
24C4×#634—22no—
25#63648—22no—
26#63643—22no—
27C22×#204—34no—
28C2×C4×Dic6—34no—
29C2×#821—34no—
30C2×#816—34no—
31#4939—34no—
32C3×#336D314yescomplemented
33#4551—34no—
34C23×C3⋊D3—21no—
35C22×C32×C4D411yescharacteristic
36C22×C32×C4D411yescharacteristic
37C22×C32×C4C2311yescharacteristic
38C22×C3⋊Dic3—21no—
39C22×C3⋊Dic3—21no—
40C22×C3⋊Dic3—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—21no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4⋊(C3⋊D3)—21no—
46C2×C4⋊(C3⋊D3)C2×C412yes—
47C2×#634—21no—
48C2×#634—21no—
49C2×#634—21no—
50C2×#634C2×C412yes—
51C2×#634D412yes—
52C2×#634D412yes—
53C2×C32×C8—21no—
54C2×C32×C8—21no—
55C2×C32×C8—21no—
56C2×C32×C8—21no—
57C2×C32×C8—21no—
58C2×C32×C8C2×C412yes—
59C32×C42—21no—
60C32×C42—21no—
61#3752—42no—
62#3752—42no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)D412yes—
67#3750—24no—
68C4×C3⋊Dic3—24no—
69#3744—28no—
70#1880—91noSylow
71C22×D12—34no—
72C22×Dic6—34no—
73C22×C3×C8C2×D314yes—
74C22×C3×C8C2×D314yes—
75C2×C3×C42C2×D314yes—
76C2×(C2×D3)⋊C4—64no—
77C2×(C2×D3)⋊C4—64no—
78C2×C4⋊Dic3—34no—
79C2×C4⋊Dic3—34no—
80C2×Dic3⋊C4—64no—
81C2×C4×Dic3—64no—
82C2×#204—316no—
83C2×#204—68no—
84C2×#204—68no—
85C4×Dic6—68no—
86C4×Dic6—68no—
87#821—68no—
88#816—68no—
89C23×C32C2×D411yescharacteristic
90C22×C3⋊D3—22no—
91C22×C3⋊D3—41no—
92C22×C3⋊D3—41no—
93C22×C3⋊D3—41no—
94C22×C3⋊D3—41no—
95C22×C3⋊D3—41no—
96C22×C3⋊D3—41no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4SD1612yescomplemented
103C2×C32×C4C22×C411yescharacteristic
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4#4111yescharacteristic
108C2×C32×C4#4111yescharacteristic
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3—41no—
114C2×C3⋊Dic3—41no—
115C2×C3⋊Dic3—41no—
116C2×C3⋊Dic3—41no—
117C2×C3⋊Dic3—41no—
118C4⋊(C3⋊D3)—22no—
119C4⋊(C3⋊D3)—22no—
120C4⋊(C3⋊D3)—41no—
121C4⋊(C3⋊D3)—41no—
122#634—22no—
123#634—22no—
124#634—22no—
125#634C22⋊C414yescomplemented
126C32×C8—22no—
127C32×C8—41no—
128C22×SD16—91no—
129C2×C4×Q8—91no—
130C2×#103—91no—
131C2×#102—91no—
132#390—91no—
133#340—91no—
134#336C3⋊D311yescharacteristic, complemented
135C23×D3—64no—
136C22×C3×C4D1214yes—
137C22×C3×C4D1214yes—
138C22×C3×C4C22×D314yes—
139C22×Dic3—64no—
140C22×Dic3—64no—
141C22×Dic3—64no—
142C2×D12—38no—
143C2×D12—64no—
144C2×D12—64no—
145C2×D12—64no—
146C2×D12—64no—
147C2×D12—64no—
148C2×Dic6—38no—
149C2×Dic6—38no—
150C2×Dic6—38no—
151C2×Dic6—64no—
152C2×Dic6—64no—
153C2×Dic6—64no—
154C2×C3×C8—24no—
155C2×C3×C8—24no—
156C2×C3×C8—24no—
157C2×C3×C8—24no—
158C2×C3×C8—24no—
159C2×C3×C8C4×D318yes—
160C3×C42—24no—
161C3×C42—24no—
162(C2×D3)⋊C4—128no—
163(C2×D3)⋊C4—128no—
164C4⋊Dic3—38no—
165C4⋊Dic3—64no—
166C4⋊Dic3—64no—
167C4⋊Dic3—64no—
168Dic3⋊C4—616no—
169C4×Dic3—616no—
170#204—632no—
171C22×C32C4×D411yescharacteristic
172C22×C32#12011yescharacteristic
173C22×C32#12111yescharacteristic
174C22×C32C2×SD1611yescharacteristic
175C22×C32#13511yescharacteristic
176C22×C32AGL(1,ℤ/8)11yescharacteristic
177C22×C32#13711yescharacteristic
178C2×C3⋊D3—42no—
179C2×C3⋊D3—42no—
180C2×C3⋊D3—42no—
181C2×C3⋊D3—81no—
182C2×C3⋊D3—81no—
183C2×C3⋊D3—81no—
184C2×C3⋊D3—81no—
185C32×C4—22no—
186C32×C4—41no—
187C32×C4C2×C22⋊C411yescharacteristic
188C32×C4C4×D411yescharacteristic
189C32×C4#12111yescharacteristic
190C32×C4#12311yescharacteristic
191C3⋊Dic3—24no—
192C3⋊Dic3—42no—
193C3⋊Dic3—81no—
194C22×Q8—91no—
195C22×D4—91no—
196C2×SD16—94no—
197C2×SD16—182no—
198C2×SD16—182no—
199C22×C8C2×C3⋊D311yescharacteristic
200C22×C8C2×C3⋊D311yescharacteristic
201C4×Q8—182no—
202C4×Q8—182no—
203C2×C4⋊C4—91no—
204C2×C4⋊C4—91no—
205C2×C4⋊C4—181no—
206C2×C22⋊C4—181no—
207C2×C22⋊C4—181no—
208C2×C42—181no—
209C2×C42C2×C3⋊D311yescharacteristic
210#103—182no—
211#102—182no—
212C23×C3C2×D1214yes—
213C22×D3—68no—
214C22×D3—124no—
215C22×D3—124no—
216C22×D3—124no—
217C22×D3—124no—
218C22×D3—124no—
219C22×D3—124no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4#20418yescomplemented
226C2×C3×C4C2×C4×D314yes—
227C2×C3×C4#23514yes—
228C2×C3×C4D3×D414yes—
229C2×C3×C4D3×D414yes—
230C2×C3×C4D3×D414yes—
231C2×C3×C4#23914yes—
232C2×Dic3—68no—
233C2×Dic3—68no—
234C2×Dic3—68no—
235C2×Dic3—68no—
236C2×Dic3—124no—
237C2×Dic3—124no—
238C2×Dic3—124no—
239C2×Dic3—124no—
240C2×Dic3—124no—
241D12—68no—
242D12—68no—
243D12—124no—
244D12—124no—
245Dic6—316no—
246Dic6—68no—
247Dic6—68no—
248Dic6—68no—
249C3×C8—28no—
250C3×C8—44no—
251C2×C32#38611yescharacteristic
252C2×C32C4×SD1611yescharacteristic
253C2×C32#44011yescharacteristic
254C2×C32#44811yescharacteristic
255C2×C32#44911yescharacteristic
256C2×C32#46011yescharacteristic
257C2×C32#46411yescharacteristic
258C3⋊D3—82no—
259C24—181no—
260C2×Q8—92no—
261C2×Q8—92no—
262C2×Q8—92no—
263C2×Q8—181no—
264C2×Q8—181no—
265C2×Q8—181no—
266C2×D4—92no—
267C2×D4—181no—
268C2×D4—181no—
269C2×D4—181no—
270C2×D4—181no—
271C2×D4—181no—
272C22×C4—181no—
273C22×C4—181no—
274C22×C4—181no—
275C22×C4C4⋊(C3⋊D3)11yescharacteristic
276C22×C4C4⋊(C3⋊D3)11yescharacteristic
277C22×C4C22×C3⋊D311yescharacteristic
278SD16—188no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C4×C3⋊D312yes—
285C4⋊C4—92no—
286C4⋊C4—184no—
287C4⋊C4—181no—
288C4⋊C4—181no—
289C4⋊C4—181no—
290C22⋊C4—362no—
291C22⋊C4—362no—
292C42—21no—
293C42—21no—
294C42—184no—
295C22×C3C4×D1214yes—
296C22×C3#88214yes—
297C22×C3#89514yes—
298C22×C3C2×#20414yes—
299C22×C3#90414yes—
300C22×C3#90814yes—
301C22×C3#90914yes—
302C2×D3—128no—
303C2×D3—128no—
304C2×D3—128no—
305C2×D3—244no—
306C2×D3—244no—
307C2×D3—244no—
308C2×D3—244no—
309C3×C4—28no—
310C3×C4—44no—
311C3×C4D3×C22⋊C414yes—
312C3×C4#88414yes—
313C3×C4#89314yes—
314C3×C4#89414yes—
315Dic3—616no—
316Dic3—128no—
317Dic3—244no—
318C32#188011yescharacteristic, complemented, Sylow
319C23—182no—
320C23—361no—
321C23—361no—
322C23—361no—
323C23—361no—
324C23—361no—
325C23—361no—
326C23C2×C4⋊(C3⋊D3)11yescharacteristic
327Q8—94no—
328Q8—182no—
329Q8—182no—
330Q8—182no—
331D4—182no—
332D4—182no—
333D4—361no—
334D4—361no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—182no—
341C2×C4—182no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4—361no—
345C2×C4—361no—
346C2×C4—361no—
347C2×C4—361no—
348C2×C4—361no—
349C2×C4#374412yescomplemented
350C2×C4C2×C4×C3⋊D311yescharacteristic
351C2×C4#382811yescharacteristic
352C2×C4D4×C3⋊D311yescharacteristic
353C2×C4D4×C3⋊D311yescharacteristic
354C2×C4D4×C3⋊D311yescharacteristic
355C2×C4#383211yescharacteristic
356C8—22no—
357C8—41no—
358C2×C3#466514yes—
359C2×C3C4×#20414yes—
360C2×C3#471014yes—
361C2×C3#473414yes—
362C2×C3#473814yes—
363C2×C3#484514yes—
364C2×C3#484814yes—
365D3—248no—
366C22—362no—
367C22—362no—
368C22—362no—
369C22—721no—
370C22—721no—
371C22—721no—
372C22—721no—
373C22C4×C4⋊(C3⋊D3)11yescharacteristic
374C22#6409111yescharacteristic
375C22#6410411yescharacteristic
376C22C2×#374411yescharacteristic
377C22#6411311yescharacteristic
378C22#6411711yescharacteristic
379C22#6411811yescharacteristic
380C4—22no—
381C4—41no—
382C4—184no—
383C4—362no—
384C4—721no—
385C4C22⋊C4×C3⋊D311yescharacteristic
386C4#6409311yescharacteristic
387C4#6410211yescharacteristic
388C4#6410311yescharacteristic
389C3#6868614yescomplemented
390C2—722no—
391C2#1059236411yescharacteristic
392C2C4×#374411yescharacteristic
393C2#1059240911yescharacteristic
394C2#1059243311yescharacteristic
395C2#1059243711yescharacteristic
396C2#1059254411yescharacteristic
397C2#1059254711yescharacteristic
subgroupG/H
centerC23C2×C4⋊(C3⋊D3)
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×#336C2
socleC23×C32C2×D4
prime radicalC22×C4⋊(C3⋊D3)C22
cosocleC22×C32×C4C23
hypercenter#336C3⋊D3
hypocenterC32#1880
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42 ◁ #336

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×#634C2semidirect
4. C2×#63643C2semidirect
6. C32×#336C2semidirect
7. #10589417C2semidirect
22. C2×#3744C4semidirect
32. C3×#336D3semidirect
102. C2×C32×C4SD16semidirect
125. #634C22⋊C4semidirect
134. #336C3⋊D3semidirect
225. C2×C3×C4#204semidirect
318. C32#1880semidirect
349. C2×C4#3744semidirect
389. C3#68686semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#37442
C2×C4×#6342
C2×#636482
C2×#636432
#105926382
C32×#3362
#105894172
#686863